Multiple absolute values with multiple variables in an LP

Question

Assume that we have a LP with the constraint $$ \sum_{j} \left(c_j x_j + |c_j x_j - \alpha_j + \beta_j|\right) \leq y $$ and

$$\alpha_j + \beta_j \leq \lambda_j $$ for all $j$, where the (positive) cost vector $c$ is known, $x_j, \alpha_j$, and $\beta_j \geq 0$ are variables. How do i linearize the absolute values, the usual trick with $|f(x)| \leq y$ would not work because we have multiple absolute values and multiple variables inside the absolute values.

Answer 1

Yes, the usual technique applies. Introduce a variable $z_j$ (together with linear constraints) to represent the absolute value, and replace the original constraint with $\sum_j (c_j x_j + z_j) \le y$.